Signals can represent information from any source that generates data, relating to electromagnetic energy to stock prices. Analysis of these signals is the focus of signal processing theory and practice. Linear prediction is an important signal processing technique that provides a number of capabilities: (1) prediction of the future of a signal from its past; (2) extraction of important features of a signal; and (3) compression of signals. The economic value of linear prediction is incalculable as its prevalence in industry is enormous.
It is observed that many important signals are “multi-channel” in that the signals are gathered from many independent sources; e.g., time series. For example, multi-channel data stem from the process of searching for oil, which requires measuring the earth at many locations simultaneously. Also, measuring the motions of walking (i.e., gait) requires simultaneously capturing the positions of many joints. Further, in a video system, a video signal is a recording of the color of every pixel on the screen at the same moment; essentially each pixel is essentially a separate “channel” of information. Linear prediction can be applied to all of the above disparate applications.
Conventional linear prediction techniques have been inadequate in the treatment of multi-channel time series, particularly, when the dimensionality is in the order is above three. There are traditional approaches of linear prediction for multi-channel signals, but are not effective in addressing the technical difficulties that are caused by the interactions of the sources of data. In single source signals, such as like voice, these difficulties are not encountered. The conventional techniques assume that the autocorrelation matrix of the data is invertible or can be made invertible by simple methods, which is rarely valid for real multi-channel data.
Also, such traditional approaches do not use the structural information available through modeling multi-dimensional geometry in a more sophisticated manner than merely as arrays of numbers. In addition, these approaches fail to take into account the phenomenon of time warping, which, for example, is critical to successful modeling of biometric time series. Further, conventional linear prediction techniques are based on a statistical foundation for linear prediction, which is not well suited for motion, video and other types of multi-channel data.
Further, it is recognized that most real multi-channel data are highly correlated. Under the conventional approaches, the popular linear prediction algorithm, known as the Levinson algorithm, cannot be applied to highly correlated channels.
Therefore, there is a need to provide a framework for extending applicability of linear prediction techniques. Additionally, there is a need for an approach to predict/compress/encrypt multi-channel multi-dimensional time series, particularly series with high correlation.